Toric Bezier patches generalize the classical tensor-product triangular and rectangular Bezier surfaces, extensively used in CAGD. The construction of toric Bezier surfaces corresponding to multi-sided convex hulls for known boundary mass-points with integer coordinates (in particular for trapezoidal and hexagonal convex hulls) is given. For these toric Bezier surfaces, we find approximate minimal surfaces obtained by extremizing the quasi-harmonic energy functional. We call these approximate minimal surfaces as the quasiharmonic toric Bezier surfaces. This is achieved by imposing the vanishing condition of gradient of the quasiharmonic functional and obtaining a set of linear constraints on the unknown inner mass-points of the toric Bezier patch for the above mentioned convex hull domains, under which they are quasi-harmonic toric Bezier patches. This gives us the solution of the Plateau toric Bezier problem for these illustrative instances for known convex hull domains.